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This paper addresses the fault detection and fault estimation issues for linear discrete time invariant (LTI) systems subject to unknown disturbances. The underlying problems are investigated based on the generalized KYP lemma in the finite frequency domain. In this paper, a new LMIs solution to the fault detection issue is proposed, which is slightly different from the available result and the set of LMIs are also simpler. A fault estimation method which focus on estimating the low frequency faults is also proposed. The fault detection and fault estimation techniques are further applied to a lab robotic manipulator.